The evolution of passive dispersal versus habitat selection have differing emergent consequences in metacommunities

Abstract

Dispersal among local communities is fundamental to the metacommunity concept but is only important to the metacommunity structure if dispersal causes distortions of species abundances away from what local ecological conditions favour. We know from much previous work that dispersal can cause such abundance distortions. However, almost all previous theoretical studies have only considered one species alone or two interacting species (e.g. competitors or predator and prey). Moreover, a systematic analysis is needed of whether different dispersal strategies (e.g. passive dispersal versus demographic habitat selection) result in different abundance distortion patterns, how these distortion patterns change with local food web structure, and how the dispersal propensities of the interacting species might evolve in response to one another. In this article, we show using computer simulations and analytical models that abundance distortions occur in simple food webs with both passive dispersal and habitat selection, but habitat selection causes larger distortions. Additionally, patterns in the evolution of dispersal propensity in interacting species are very different for these two dispersal strategies. This study identifies that the dispersal strategies employed by interacting species critically shape how dispersal will influence metacommunity structure.

This article is part of the theme issue ‘Diversity-dependence of dispersal: interspecific interactions determine spatial dynamics’.

1. Introduction

All organisms face a basic dilemma at one or more life stages—‘Do I stay or do I go?’ (with nods to The Clash). Many organisms make ‘decisions’ as to whether fitness is best served by remaining in place or moving elsewhere. In some cases, the ‘decision’ is not made by the dispersing organism, but by its parent: e.g. whether or not a plant seed has an awn facilitating dispersal depends on resource allocation by the maternal plant [1,2]. Often, dispersal is restricted to a single life history stage (e.g. seeds of plants, larvae of space-occupying organisms in the intertidal zone such as barnacles), but in some species, dispersal can occur at many points across the life cycle (e.g. humans). Two distinct aspects of dispersal are often conflated: whether to disperse at all (emigration) and where to settle if one emigrates (immigration). These are different processes, and each can be under selection [3,4]. This holds even for passively dispersed organisms that have some control over the dispersal process (writ large)—a female may choose the conditions under which to release propagules [5], and released propagules may choose under what conditions they will settle [6]. So, from highly mobile animals (e.g. birds) to sessile marine invertebrates with dispersing larvae, organisms often exert some influence over where they or their offspring attempt to establish long term: i.e. demographic habitat selection [7,8] (hereafter simply habitat selection) in contrast to habitat selection in the optimal foraging sense [9]. Yet, many other organisms experience almost entirely passive dispersal, both immigration and emigration: e.g. microbes, planktonic species and plants with tiny seeds and bodies transported by strong physical processes such as the wind and water currents [10,11]. In such species, emigrants exert little or no control over the direction of dispersal or their resulting settlement. In either case, natural selection can shape both the propensity to emigrate at all, and the algorithms controlling that ‘decision’, and for many motile organisms the algorithms controlling where to immigrate as well.

A prime driver of selection on dispersal is spatiotemporal variation in fitness (others include reducing competition with kin and avoiding inbreeding) [7,12–17]. Fitness is determined by both the local density of the focal species, given density dependence, and the abundances of interacting species such as predators, prey, competitors, pathogens and mutualists. Temporal variation in the environment can directly cause spatiotemporal variation in fitness (e.g. due to weather fluctuations) or be an indirect cause (by altering local densities of conspecifics and heterospecifics). Dispersal is the central process linking communities into metacommunities [18,19], but the metacommunity literature largely ignores the distinction between emigration and immigration and the potential consequences of passive dispersal versus active habitat selection (but see [20–22]). Emigration may be modelled as a proportion of the existing population in a patch that leaves, influenced by a variety of factors (e.g. density, environmental change and predation risk), but the realized immigration rate for organisms that can evaluate relative expected fitnesses of available habitat patches, and chose accordingly, likely varies among patches.

We here explore how dispersal evolution in a community context might differ between passive dispersers and active habitat selectors, and how such evolution influences emergent patterns of species’ abundances in metacommunities. We use models to explore how simple communities behave under a variety of dispersal rates and contrast two specific dispersal processes—passive dispersal and habitat selection—across a metacommunity. Our goal is to lay the mechanistic groundwork for assessing how species-level variation in dispersal strategies influences both the evolution of dispersal in complex landscapes and the assembly of communities and metacommunities at meaningful spatial scales. We analyse how evolution in passive dispersal versus habitat selection influences species abundances across a landscape containing multiple patches each of which harbours a diamond/keystone predator food web of four interacting species and various subsets of this community. We focus on this simple food web structure to illustrate both how different dispersal strategies differentially affect patterns in abundance, and how complex and unexpected results can arise, even in simple food webs.

2. A multispecies model with the evolution of dispersal propensity

In this section, we develop a discrete-time simulation model that includes the abundance dynamics and coexistence of species in the classic diamond/keystone predation food web of one basal prey, two intermediate consumers and one top predator in one local community, a scenario that has been thoroughly explored [23–25]. We use this model to explore how the evolution of dispersal propensity and abundance distortions may shape the metacommunity structure for a food web of interacting species. To our knowledge, there has been no exploration about how this suite of interactions modulates the evolution of dispersal, nor how such evolution in turn impacts overall abundances across the food web.

We consider a metacommunity of 16 patches ( $g=1,2,\mathrm{...},16$ ) in a landscape with this module distributed throughout. (We have performed analyses for metacommunities containing 2–30 patches and found qualitatively identical results across patch numbers. Therefore, we only present results for 16-patch metacommunities.) The basal species’ ( $R_{\left(g\right)}$ ) population dynamics follow logistic population growth, where $c_{\left(g\right)}\left(t\right)$ is its local intrinsic rate of increase and $d$ is the strength of density dependence. This species is fed upon by two consumer species at the next higher trophic level ( $N_{k\left(g\right)}$ with $k=1,2$ ). These consumers have linear functional responses, with $a_k$ attack coefficients and $b_k$ conversion efficiencies. These two consumer species are themselves fed upon by a single predator species at the third trophic level ( $P_{\left(g\right)}$ ), also with linear functional responses for feeding on the two consumers, with $m_k$ attack coefficients and $n_k$ conversion efficiencies. The intermediate trophic level consumers and top predators have intrinsic death rates of $f_{k\left(g\right)}\left(t\right)$ and $x_{\left(g\right)}\left(t\right)$ , respectively. We assume that the attack coefficients and conversion efficiencies of all species are invariant, but the intrinsic rates of increase for $R_{\left(g\right)}$ , and the intrinsic death rates of $N_{1\left(g\right)}$ , $N_{2\left(g\right)}$ and $P_{\left(g\right)}$ may vary in time and space. The dynamics in each local community (without dispersal) are described by a system of difference equations (a multispecies generalization of the Ricker single-species model):

$${\displaystyle {\displaystyle \begin{array}{c}{P}_{\left(g\right)}\left(t+1\right)=P_{\left(g\right)}\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\sum _{k=1}^2{n}_k{m}_k{N}_{k\left(g\right)}\left(t\right)-x_{\left(g\right)}\left(t\right)\right)\\ N_{k\left(g\right)}\left(t+1\right)=N_{k\left(g\right)}\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(b_k{a}_k{R}_{\left(g\right)}\left(t\right)-m_k{P}_{\left(g\right)}\left(t\right)-f_{k\left(g\right)}\left(t\right)\right)\\ R_{\left(g\right)}\left(t+1\right)=R_{\left(g\right)}\left(t\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(c_{\left(g\right)}\left(t\right)-d{R}_{\left(g\right)}\left(t\right)-\sum _{k=1}^2{a}_k{N}_{k\left(g\right)}\left(t\right)\right)\end{array}.}}$$ (2.1)

Levins demonstrated that temporal variation in density-independent demographic rates alone with no dispersal does not distort the long-term time-averaged abundances of interacting species [26], and our simulations of this specific model with no dispersal confirm these conclusions. Coexistence conditions likewise are not changed when constant density-independent parameters are replaced by their time averages.

Following population regulation, species disperse among patches. Each species harbours a number of asexual dispersal genotypes, each with their own dispersal propensities, and which are otherwise ecologically identical. Consequently, local population regulation is based on total local population size, and regulation alone does not change the relative proportions of dispersal types within a patch. Dispersal rate evolves in each species only because of differences among these asexual genotypes in movement among patches in the metacommunity, and how dispersal averages across spatiotemporal variation in fitness.

We consider two forms of dispersal in this analysis: passive dispersal and adaptive habitat selection. With passive dispersal, each dispersal type has a specified dispersal propensity ${\epsilon }_{\left(q\right)}$ , the percentage of individuals of dispersal type q emigrating from their natal patch each generation. These dispersers are then apportioned equally among (i.e. immigrate into) all the other patches in the metacommunity. Mathematically for dispersal type q of species X, this is expressed as:

$${\displaystyle {\displaystyle X_{\left(g\right)}^{\left(q\right)}\left(t+2\right)=X_{\left(g\right)}^{\left(q\right)}\left(t+1\right)-{\epsilon }^{\left(q\right)}{X}_{\left(g\right)}^{\left(q\right)}\left(t+1\right)+\sum _{j=1\left(g\ne j\right)}^n{\epsilon }^{\left(q\right)}\frac{X_{\left(j\right)}^{\left(q\right)}\left(t+1\right)}{\left(n-1\right)},}}$$ (2.2)

where time t + 1 is after regulation but before dispersal and n is the number of patches in the metacommunity. Thus, one complete iteration of a simulation involves these two time steps. For the results presented here, each species initially has 21 dispersal types, with dispersal propensities ${\epsilon }_{\left(q\right)}$ evenly spaced at an interval of 0.05 from 0 to 1. Only one species directly experiences temporal variation in its demographic parameters, but all species can potentially evolve in their dispersal propensities.

For habitat selection, species also comprise 21 dispersal types with dispersal propensities arrayed from 0 to 1 at 0.05 intervals. Dispersal propensity is the probability of emigrating from the natal patch. To be clear, we are modelling habitat matching demographic habitat selection, where organisms match their phenotype to characteristics of the habitat patch affecting fitness [27]. Thus, after population regulation but before dispersal, the per capita fitness for each species in each patch (exponential terms in equation (2.1)) is recorded based on current abundances. These are the fitness values individuals use to assess two patches—their natal patch and the patch to which they have dispersed. Dispersing individuals are initially equally apportioned to all other patches in the metacommunity. However, each group of potential immigrant individuals then compares the fitness value for the new patch to the fitness value of their natal patch (fitness is given by the exponential terms on the right-hand side of equation (2.1), using the abundances after regulation but before dispersal). If the fitness value of the new patch is higher, they immigrate to the new patch. If the assessed fitness value of the natal patch is higher, they return to their natal patch. We note that this behaviour is adaptive, because given density dependence within and among species, and the fact that local abundances have short-term positive autocorrelations, there is likewise short-term predictability in fitness (albeit not in its density-independent component). However, this behaviour is not optimal; the latter would require omniscience of both all potential fitness returns in all patches and the dispersal choices of all other individuals in the metacommunity.

Denoting the per capita fitness value in patch i (i.e. the corresponding exponential term in equation (2.1) using abundances after regulation) as $W_{\left(i\right)}\left(t+1\right)$ , this dispersal strategy can be expressed mathematically as:

$${\displaystyle {\displaystyle X_{\left(g\right)}^{\left(q\right)}\left(t+2\right)=X_{\left(g\right)}^{\left(q\right)}\left(t+1\right)-\sum _{i=1\left(g\ne i\right)}^n{E}_{\left(g,i\right)}^{\left(q\right)}\left(t+1\right)\frac{X_{\left(g\right)}^{\left(q\right)}\left(t+1\right)}{n-1}+\sum _{j=1\left(g\ne j\right)}^n{I}_{\left(j,g\right)}^{\left(q\right)}\left(t+1\right)\frac{X_{\left(j\right)}^{\left(q\right)}\left(t+1\right)}{\left(n-1\right)},}}$$ (2.3)

$${\displaystyle \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}{E}_{\left(g,i\right)}^{\left(q\right)}\left(t+1\right)=\begin{array}{cc}0& \text{if}{W}_{\left(g\right)}\left(t\right){>}_{\left(i\right)}\left(t\right)\\ {\epsilon }^{\left(q\right)}& \text{if}{W}_{\left(g\right)}\left(t\right){<}_{\left(i\right)}\left(t\right)\end{array}\mathrm{a}\mathrm{n}\mathrm{d}{I}_{\left(j,g\right)}^{\left(q\right)}\left(t+1\right)=\begin{array}{l}\{\begin{array}{ll}{\epsilon }^{\left(q\right)}& \text{if}{W}_{\left(g\right)}\left(t\right)>W_{\left(j\right)}\left(t\right)\\ 0& \text{if}{W}_{\left(g\right)}\left(t\right)<W_{\left(j\right)}\left(t\right)\end{array}\end{array}.}$$

Because this is a population-level and not an individual-based model, individuals do not assess fitness; the entire cohort of dispersers from one specific patch to another specific patch assess the fitness differences together. Also, because all are moving simultaneously, there is no ordering by which dispersers from different patches can assess according to the choices of other dispersers. We felt that this was the best compromise—all dispersers of all the species to all patches evaluate fitness based on fitness before this mass movement occurs. Therefore, the language of ideal-free distributions or optimal dispersal does not apply. This movement strategy is adaptive, but not necessarily optimal (e.g. the natal patch might not have the highest fitness across all patches in the metacommunity, and a disperser’s realized fitness in the next iteration will also depend on the choices of all other dispersers). This is simply one of an immense set of strategies one could explore where dispersers are choosing habitats based on some fitness criterion. For simplicity, we assume cost-free dispersal.

To characterize the final evolutionarily stable dispersal propensity, the average dispersal propensity was calculated among the dispersal types based on their frequencies at the end of a simulation run. Typically, one or two adjacent dispersal types were at very high frequency and all others at extremely low frequencies (i.e. <10⁻⁵). For such a single-peaked frequency distribution, this average was taken to be the dispersal propensity favoured by natural selection. In some cases, the frequency distribution was strongly bimodal, indicating a protected polymorphism of two dispersal propensities was favoured. In this latter case, an average was taken around each peak, with the break between the two peaks taken to be the dispersal propensity with the lowest frequency between them. Finally, with passive dispersal, the frequencies of dispersal types of many species changed very little over a simulation. For these species, we interpreted this as little to no evolution in dispersal propensity.

It has long been recognized that the evolution of dispersal propensity is fostered by spatiotemporal variation in fitness among patches [14,15,17,28–30]. Spatiotemporal variation was generated by making one density-independent parameter for a single species in any given simulation run a random variable. We explored the consequences of spatiotemporal variation in the basal resource’s intrinsic rate of increase $c_{\left(g\right)}\left(t\right)$ , the consumers’ intrinsic death rates $f_{k\left(g\right)}\left(t\right)$ or the predator’s intrinsic death rate $x_{\left(g\right)}\left(t\right)$ . Demographic variation for one species may impact others in the metacommunity because of variation generated in that species’ abundance, which could cause fitness variation across the web of interacting species. For each iteration of a simulation, values for the variable parameter were randomly drawn from a normal distribution with specified mean and standard deviation (s.d.); a separate value was chosen for each patch at each iteration to generate spatiotemporal variation. We assumed no covariation in parameters in space or time. We implemented this model in Java v.8 (Oracle Corp.). The code is available in the electronic supplementary materials.

For our analyses, we used species interactions characterized by one set of parameter values as illustrated in figure 1. These parameter values imply that $N_1$ is the better resource competitor for $R$ but suffers higher predation rate from $P$ than does $N_2$ . These differences between $N_1$ and $N_2$ permit them to coexist if both $R$ and $P$ are present with no spatial or temporal variation [23–25,31]. In simulations, the following standard deviations were used: for $c_{\left(g\right)}\left(t\right)$ a s.d. of 0.5; for $f_{k\left(g\right)}\left(t\right)$ a s.d. of 0.1 and with $x_{\left(g\right)}\left(t\right)$ a s.d. of 0.025 (these choices of parameter distributions make it very unlikely that biologically unrealistic values will occur during simulation runs). Extensive simulations using other parameter combinations where $N_1$ and $N_2$ coexist show that the results presented here represent general features of environmental variation and dispersal on the ecological and evolutionary dynamics of interacting species in a metacommunity (details not shown). We present this example to illustrate the diversity of outcomes possible in a metacommunity with different species combinations and different dispersal strategies. Each simulation was run for 50 000 iterations and 100 replicate simulations were run for each species combination and dispersal strategy. Standard errors (s.e.) among replicates for average abundances and dispersal frequencies were generally two orders of magnitude lower than mean values, so only the mean values among the replicates are presented.

Figure 1.

Full four species, three trophic level food web and associated parameters used in the simulation studies of the effects of temporal environmental variation and dispersal on food web structure. All simulations were performed in a metacommunity with 16 patches. The mean parameter values for all patches are given next to the arrows in the diagram. The arrow pointing from no species to R gives the values of the logistic growth function ( $c_{\left(g\right)}$ and $d_{\left(g\right)}$ ). The arrows pointing from no species to the other three species give the values of the intrinsic death rates ( $f_{k\left(g\right)}$ and $x_{\left(g\right)}$ ) for those species. The arrows pointing from one species to another give the values of the attack coefficients for the species at the higher trophic level consuming the species at the lower trophic level ( $a_{k\left(g\right)}$ and $m_{k\left(g\right)}$ ). The conversion efficiencies of the consumers and predators $\left(b_k,n_k\right)$ are all 0.1.

3. Evolution of dispersal propensity

(a). Passive dispersal

With passive dispersal, the dispersal propensity of only the species experiencing variation in its own specific demographic parameter evolved in response to spatiotemporal variation. With variation in its intrinsic rate of increase $c_{\left(g\right)}\left(t\right)$ , $R$ evolved a propensity equal to the theoretical expectation of 15/16 = 0.9375 (i.e. number of patches − 1/number of patches) (table 1a ) (see appendix A for a derivation of this result for the optimal passive dispersal propensity). With variation in its intrinsic death rate $f_{k\left(g\right)}\left(t\right)$ , $N_k$ evolved a somewhat lower dispersal propensity around 0.80–0.83 (table 1a ). However, variation in $x_{\left(g\right)}\left(t\right)$ , the predator’s intrinsic death rate, induced no evolution in its dispersal propensity with passive dispersal (table 1a ). Intriguingly, dispersal evolves differently at different trophic levels due to the same kind of temporal variation in a demographic rate. Moreover, dispersal propensity only evolved in the single species that directly experienced variation in their parameter. These results illustrate that community context, the source of temporal variation and the position of a species in a food web, can all influence the outcome of dispersal evolution. Variation directly impacting one species’ fitness (as assumed in our model) is expected to lead to temporal variation in abundances and fitnesses across the food web. Intriguingly, such variation does not necessarily generate selection on dispersal.

Table 1.

Dispersal propensities that evolve in response to spatiotemporal variation in a population parameter of one species. The first column identifies the stochastic parameter for the rows of results. In the last column (i.e. P–N ₁–N ₂–R), two sets of results are given for variation in f for N _i : the first set for variation in f ₁ and the second set for variation in f ₂. For each set, responses for P, N _i and R are given in rows. Column headers identify the community module used in simulations. N ₂–R results are essentially the same as N ₁–R, and thus excluded for brevity. Species either evolved to a single dispersal type identified by a single number here or a polymorphism of two dispersal types given by two numbers in square brackets with the frequencies of the two given in parentheses after them. ‘no selection’ means that the frequencies of all dispersal types had changed little over 50 000 iterations. The numbers are averages across 100 replicates for each community module. The s.e. across those replicates are all on the order of 0.01 or much smaller (results not shown).

		(a) passive dispersal
		R	N 1-R	P-N 1-R	P-N 2-R	P-N 1-N 2-R
variance in c for R	P	—	—	no selection	no selection		no selection
	N i	—	no selection	no selection	no selection		no selection	no selection
	R	0.94	0.94	0.94	0.94		0.94
variance in f for N	P		—	no selection	extinct	no selection		no selection
	N i		0.83	0.82	0.82	0.80	no selection	no selection	0.80
	R		no selection	no selection	no selection	no selection		no selection
variance in x for P	P			no selection	extinct		no selection
	N i			no selection	no selection		no selection	no selection
	R			no selection	no selection		no selection

		(b) habitat selection
		R	N 1-R	P-N 1-R	P-N 2-R	P-N 1-N 2-R
variance in c for R	P	—	—	1	[0,1], (0.02,0.98)		1
	N i	—	0	0.02	[0,1], (0.98,0.02)		0.05	0.04
	R	0.53	0.84	0.66	0.77		0.75
variance in f for N	P		—	0	0	0		0
	N i		1	1	1	1	1	1	1
	R		[0.15,1.0]	[0.37,1.0]	[0.23,1.0]	[0.35,1.0]		[0.23,1.0]
			(0.60,0.40)	(0.77,0.23)	(0.75,0.25)	(0.83,0.17)		(0.74,0.26)
variance in x for P	P			1	1		1
	N i			1	1		1	1
	R			[0.24,0.97]	[0.24,0.89]		[0.25,0.90]
				(0.56,0.44)	(0.45,0.55)		(0.44,0.56)

(b). Habitat selection

In striking contrast, when all species in the metacommunity were habitat selectors, the dispersal propensities of all species evolved in response to spatiotemporal variation in any one of the specific parameters (table 1b ). With variation in $c_{\left(g\right)}\left(t\right)$ , $R$ evolved a moderate dispersal propensity (0.53–0.84) depending on what other species are present, in all the metacommunities considered. In response to variation in $c_{\left(g\right)}\left(t\right)$ , $N_k$ evolved a very low dispersal propensity (0.00–0.05 depending on which other species are present) and $P$ evolved a very high dispersal propensity (=1.0, always attempt dispersal) in metacommunities with only $N_1$ or with both consumers present (table 1b ). However, in the metacommunity with just $N_2$ at the intermediate trophic level, variation in $c_{\left(g\right)}\left(t\right)$ caused both $N_2$ and $P$ to evolve dispersal propensity polymorphisms with one type that does not disperse and another that attempts dispersal via habitat selection every iteration. In these polymorphisms, the no-dispersal type for $N_2$ is at a frequency of 0.98 and the no-dispersal type for $P$ is at a frequency of 0.02 (table 1b ).

Variation in either $f_{k\left(g\right)}\left(t\right)$ for $N_k$ or $x_{\left(g\right)}\left(t\right)$ for $P$ caused $R$ to evolve a polymorphism of moderate and high dispersal types in all metacommunities. Variation in $f_{k\left(g\right)}\left(t\right)$ resulted in $R$ evolving a polymorphism having moderate dispersal type with propensity of 0.15–0.37, and an always-attempt-dispersal type with low dispersal type at higher relative frequency (table 1b ). Variation in ${\displaystyle x\left(g\right)\hspace{0.17em}\left(t\right)}$ resulted in a polymorphism for $R$ having moderate dispersal type with propensity of 0.24–0.25 and high dispersal type with propensity of 0.89–0.97, with the relative frequencies depending on the species present (table 1b ). Variation in $f_{k\left(g\right)}\left(t\right)$ caused $N_k$ to evolve a dispersal propensity of 1.0 and $P$ a dispersal propensity of 0.0 in all metacommunities. Variation in $x_{\left(g\right)}\left(t\right)$ for $P$ caused both $N_k$ and $P$ to evolve propensities of 1.0 in all metacommunities. Again, these results show the great importance of community context for dispersal evolution.

4. Propagation of abundance distortions in food webs

Abundance distortions because of spatiotemporal variation in fitness coupled with dispersal is an additional cause generating variation in community structure across patches. In appendix B, we derive a model for a single species that passively disperses among patches while experiencing spatiotemporal variation in its intrinsic rates of increase across the patches. By applying a Taylor series expansion, we derive the species’ long-term per capita growth rates in the patches and show that such spatiotemporal variation coupled with dispersal among patches causes an inflation in the abundance of the species in all patches in the metacommunity. The magnitude of the abundance inflation increases with both the magnitude of spatiotemporal variation in abundances and the dispersal propensity. These same abundance inflations occurred in our discrete–time simulations. In this section, we describe how the abundance inflation of the species experiencing spatiotemporal parameter variation was transmitted through the rest of the food web in the simulation results we discussed in the previous section.

Passive dispersal and habitat selection created similar patterns of abundance distortion across these simple food webs (table 2). First, consider spatiotemporal variation in $c_{\left(g\right)}\left(t\right)$ starting with the metapopulation with only $R$ present. The long-term average abundance of $R$ across all 16 patches is greater with spatiotemporal variation in $c_{\left(g\right)}\left(t\right)$ and dispersal than with no variation in $c_{\left(g\right)}\left(t\right)$ (i.e. $c_{\left(g\right)}\left(t\right)=c$ ) and no dispersal (table 2). Introducing $N_1$ (or N ₂) eliminates this inflation of $R$ ’s abundance because it is now controlled by $N_1$ (N ₂) and the abundance inflation is transferred instead to $N_1$ (N ₂) (figure 2a ). In turn, introducing $P$ eliminates $N_1$ ’s abundance inflation because $P$ controls $N_1$ ’s abundance, and so both $R$ and $P$ are now inflated in abundance (table 2 and figure 3a ). $N_1$ ’s abundance is the same as with no variation if species are passively dispersing, but it is significantly depressed if species use habitat selection, while both R and P are again inflated in abundance (figure 3a ). However, if $N_2$ is introduced instead of $N_1$ , spatiotemporal variation in $c_{\left(g\right)}\left(t\right)$ inflates $N_2$ ’s abundance enough to permit $P$ to persist, whereas $P$ is unable to support a population on $N_2$ in a static environment without this inflationary effect (figure 3d ). With all four species present, variation in $c_{\left(g\right)}\left(t\right)$ , with either passive dispersal or habitat selection, shifts the relative abundances of the two consumers in favour of $N_2$ , the species that is the poorer resource competitor but experiences less predator-inflicted mortality, but does not inflate $P$ (table 2 and figure 4a ).

Table 2.

Abundance distortions caused by spatiotemporal environmental variation and dispersal type. Numbers give the ratio of the average over 100 replicates of the abundance of the species with spatiotemporal variation to its abundance with no spatiotemporal variation. Thus, a value of 1.000 means the species abundance was the same in both conditions. Numbers given in italic are those where the abundance was significantly increased and in bold are those where abundance was significantly decreased. Table is formatted as in table 1. Statistical significance (at α = 0.05) was evaluated by determining whether the 95% confidence interval included 1.0. Again, N ₂–R results are essentially the same as N ₁–R, and thus excluded for brevity.

		(a) passive dispersal
		R	N 1-R	P-N 1-R	P-N 2-R	P-N 1-N 2-R
variance in c for R	P	—	—	1.278	extinct to present		1.000
	N i	—	1.097	1.000	1.013		0.461	1.911
	R	1.058	1.000	1.096	1.115		1.000
variance in f for N	P		—	1.034	extinct	1.153		0.812
	N i		1.012	1.000	1.015	1.142	0.760	0.749	1.424
	R		0.982	1.000	0.983	1.028		0.950
variance in x for P	P			1.002	extinct		0.990
	N i			0.999	1.000		0.987	1.021
	R			1.001	1.001		0.998

		( b ) habitat selection
		R	N 1-R	P-N 1-R	P-N 2-R	P-N 1-N 2-R
variance in c for R	P	—	—	1.225	extinct to present		1.002
	N i	—	1.075	0.940	0.942		0.397	1.836
	R	1.023	1.000	1.080	1.148		1.004
variance in f for N	P		—	1.571	extinct to present	1.964		1.216
	N i		1.196	1.000	1.013	1.232	0.609	0.000	2.689
	R		0.719	1.021	0.994	1.062		0.810
variance in x for P	P			1.335	extinct to present		1.027
	N i			0.953	0.951		0.035	2.467
	R			1.056	1.076		0.883

Figure 2.

Results of variation in (a) c for R and (b) f ₁ for N ₁ in communities consisting of R and N ₁, when the species passively disperse or exercise habitat selection. N ₂–R results are essentially the same as N ₁–R, and thus excluded for brevity. The top panel gives the average abundances of the two species in parentheses when they experience spatiotemporal variation in either c or f ₁ but do not disperse (which is also the same abundances with no spatiotemporal variation). For the results in (a), c in each patch had a s.d. of 0.5 and f ₁ had s.d. = 0.1 for the results given in (b). In parentheses after the species identifier in each food web depiction, the number to the left of the colon is the mean abundance of that species with that variation source and dispersal type, and the numbers to the right of the colon are the evolved dispersal propensities. If a single number is given, it is the single dispersal propensity that evolved. If two numbers are given in square brackets, then numbers are the dispersal propensities of the polymorphism of two dispersal propensities that evolved. If the dispersal propensity is given in red, dispersal propensity was a neutral trait that evolved very little (i.e. all 21 dispersal propensity types were at relatively high frequency at the end of simulations). All these numbers are averages of 100 replicate simulations for each condition, and the s.e. for each mean is on the order of 0.01 or smaller, and so are not presented.

Figure 3.

Results of variation in (a) c for R, (b) f ₁ for N ₁ and (c) x for P in communities consisting of R, N ₁ and P and variation in (d) c for R, (e) f ₂ for N ₂ and (f) x for P in communities consisting of R, N ₂ and P when the species passively disperse or exercise habitat selection. The presentation in this figure is exactly as organized in figure 2. No evolved dispersal propensity (i.e. ---) is given for P when it went extinct in simulations.

Figure 4.

Results of variation in (a) c for R, (b) f ₁ for N ₁, (c) f ₂ for N ₂ and (d) x for P in communities consisting of R, N ₁, N ₂ and P when the species passively disperse or exercise habitat selection. The presentation in this figure is exactly as organized in figure 2.

If either $N_1$ or N ₂ alone is present with $R$ , spatiotemporal variation in $f_{k\left(g\right)}\left(t\right)$ inflates the consumer’s abundance, which in turn depresses $R$ ’s abundance (figure 2). Again, because $P$ controls both consumer’s abundance, adding $P$ to either of these metacommunities causes an inflation of $P$ ’s abundance (again, $P$ cannot coexist feeding only on $N_2$ without spatiotemporal variation) and eliminates the inflation of the consumer’s abundance (figure 3b,e ). When all four species are present, spatiotemporal variation in $f_{k\left(g\right)}\left(t\right)$ for either species inflates that consumer’s abundance and decreases the other consumer’s abundance (figure 4b,c ). Additionally, if the variation is in $N_1$ , $R$ ’s abundance increases, and if the variation is in $N_2$ , $R$ ’s abundance decreases (figure 4b,c ). Whether $P$ ’s abundance is increased or decreased depends both on which consumer is experiencing the spatiotemporal variation and the dispersal strategies being used (table 2 and figure 4b,c ).

The biggest differences in responses between the two dispersal strategies are evident for spatiotemporal variation in $x_{\left(g\right)}\left(t\right)$ for $P$ . With passive dispersal, spatiotemporal variation in $x_{\left(g\right)}\left(t\right)$ has no inflationary effect on any species in food chains (figure 3f ). In contrast, if all species are habitat selectors, spatiotemporal variation in $x_{\left(g\right)}\left(t\right)$ causes an increase in $P$ ’s abundance in all community modules, although the effect is small in the keystone module (figure 3 and 4). Consequently, in the food chains, the intermediate consumer’s abundance is decreased, and $R$ ’s abundance is increased (table 2b and figure 3c ). In the keystone predation module, inflation of $P$ ’s abundance caused a much more dramatic shift in consumer’s relative abundances in favour of N ₂, which in turn also caused a decrease in $R$ ’s abundance (figure 4d ). The result is that in a simple four-species food web, the metacommunities produced by variation in dispersal strategy between passive dispersal and habitat selection are strikingly different.

5. Discussion

The central issue that makes the concept of a metacommunity relevant is whether dispersal among local communities distorts the distributions and abundances of component species away from what local ecological dynamics favour in the absence of dispersal [18,32,33]. Prior studies of single species help illuminate when such distortions might arise. First, in a temporally constant environment, passive dispersal is either neutral (if it is cost-free and the environment is spatially homogeneous) or actively selected against [16,34,35], if abundances are large enough that one can ignore finite population size effects (e.g. competition among kin and inbreeding depression). Ideal-free habitat selection behaviour in constant environments leads to communities where local abundances reflect just local processes [36]. Temporal variability drives dispersal evolution, and thus makes a metacommunity more than the simple sum of its community parts. Analyses of single-species dynamics in spatiotemporally variable environments have also revealed an intriguing emergent property of the interplay of dispersal and temporal variation, which has been dubbed the ‘inflationary effect’. Theoretical and experimental studies of source–sink dynamics have shown that temporal variation in local growth rates in sink habitats can elevate the long-term average abundance of the sink habitat, sometimes to a large extent [37]. This inflationary effect carries over to environments where this is no defined source, but all local habitats on average are sinks (this has been demonstrated theoretically [38] and empirically [39]). Expression (4) in Roy et al. [38] shows that the overall growth rate can be partitioned into two components: (i) a spatial average of local growth rates and (ii) a term for the covariance of local growth and densities. When the latter is positive, the former must be negative at equilibrium, which in general implies that local abundances are elevated above their long-term average in the absence of dispersal.

What we have done here is consider how the evolution of dispersal on the one hand, and the existence and expression of the inflationary effect on the other, are modulated when species are embedded in communities of interacting species. We have done this for two alternative modalities of dispersal—passive dispersal, and a form of adaptive habitat selection. In general, temporal variation in local growth rates does lead to dispersal evolution, and the inflationary effect is robust to the presence of multispecies interactions. However, we have come across some intriguing phenomena, which we attempt to address in the next few paragraphs. We have an intuitive understanding in some cases of our outcomes, but in others, our intuition fails us, and further theoretical analyses will be required to grasp fully the causal drivers of our results.

Our results illustrate two important features that strongly affect metacommunity dynamics. First, the propensity and pattern of movement among local communities favoured by natural selection depend critically on what dispersal strategies are employed by interacting species. Also, dispersal in the context of spatiotemporal variation in demographic conditions for one species can inflate its abundance in all local communities (as has been shown in analyses of single-species metapopulation dynamics) and in turn alter abundances of other species throughout the local food webs. The pattern of this distortion depends on both the structure of the food webs and the dispersal strategies used by species to move among local communities.

Some species disperse passively among communities, meaning that dispersers randomly move among local communities and do not decide where to immigrate based on any cue about the local conditions in a community they enter, e.g. wind-dispersed seeds have no influence on where they settle [5,40,41]. Dispersers of other species may exercise a modicum of choice, with rather crude discrimination; they may settle in broadly acceptable habitat (viable) and avoid unacceptable (inviable) habitat, but not make further distinctions: many dispersing insects may only settle in specific types of environment (e.g. aquatic insects only settle in ponds) but do not exercise any finer discrimination within that category [42]. In contrast, true habitat selectors discriminate among viable habitats based on any number of axes of presumed ‘patch quality’ (that collectively govern expected fitness), responding to variation in such factors as predators, prey, competitors, density, patch size, productivity, etc. [43–48]. Variation among species in the factors affecting patch quality can generate species sorting within metacommunities [49], while sharing of patch preferences among species can generate positive covariances in abundance, intensifying local species interactions (habitat compression [20]) within metacommunities.

One surprising result is that if only one species experiences temporal variation in a density-independent growth rate and uses a passive dispersal strategy, it evolves in dispersal propensity, but this does not generate selection for dispersal propensity evolution in species with which it directly interacts. For passive dispersers, the propensity to move among habitats that on average are all the same, but experience temporal variation in fitness components, evolves to minimize variation in fitness among local communities at a given time in the face of spatiotemporal fitness variation, and thus maximize long-term average fitness [14,17,50,51]. Consequently, the evolution of passive dispersal decreases the variance in abundance among local communities because of spatial averaging [52]. Thus, passive dispersal in one species does not generate fitness variation for other species with which it directly interacts in the metacommunity but which are not affected by the same environmental features that generate fitness variation in the evolving species. The optimal level of passive dispersal that evolves causes the abundances of the species experiencing environmental variation to be equal across all patches immediately after dispersal, and consequently interacting species experience no spatial variation. This is why dispersal propensity only evolved in the species directly experiencing variation in its own parameter values in the simulations with passive dispersal (table 1a ).

With habitat selection, the evolution of dispersal propensity in one species by contrast caused the evolution of dispersal propensities in all species in the food web (table 1b ). For the form of habitat selection, we have assumed a dispersing individual tests whether a patch has higher fitness than its natal patch, and so it may ultimately not disperse. Consistent patterns emerged for evolutionary responses in species at different trophic levels. Species that directly experienced spatiotemporal fitness variation always evolved a high propensity for dispersal, either a high propensity for the species at the bottom of the food web or a propensity of 1.0 if the species were a predator of others (table 1b ). Alternatively, predators of the species directly experiencing the spatiotemporal fitness variation evolved a very low or zero dispersal propensity (table 1b ). The causes for these intriguing emergent and divergent evolutionary patterns are unclear.

Dispersal among local communities can also distort the abundances of species away from values produced by local ecological conditions—the mass effect perspective for metacommunities [18]—but only under specific conditions. If a species’ abundance is regulated to different values in different local communities, dispersal among them will make its abundance in the various patches more similar, because high-abundance patches are net exporters and low-abundance patches are net importers [16]. In the extreme, such movement is the cause of source–sink structures in which a species is maintained in a local community by immigration where it could not otherwise support a population [53–55]. Additionally, when a species is regulated to different abundances in various local communities, passive dispersal among the patches can either increase or decrease its total abundance across all habitat patches, depending on how density dependence varies across space ([56–59, 60] and see appendix A). Temporal variability can also alter total metapopulation abundance [38]. However, if individuals employ a ‘balanced’ passive dispersal strategy in which they have high dispersal propensities from low-abundance patches and low dispersal propensities from high-abundance patches [17,34,50,61,62], or if individuals employ a habitat selection strategy resulting in an ideal-free distribution among patches [7,16,36,63–65], the net movement of individuals among patches will not cause local abundance distortions. In the balanced dispersal case, this is driven by the inverse relationship between abundance and dispersal (emigration) at the optimal dispersal propensity, while in the case of the ideal-free distribution, lack of local abundance distortions arises from the consistent pattern of colonization (immigration) in response to spatiotemporal variation in patch quality.

In contrast to these prior results in the literature, the simple form of habitat selection we have simulated here causes more frequent and larger overall abundance distortions than those seen with passive dispersal, thus the relative abundance structure of the communities assembled are quite different. This is especially evident when looking at the four-species keystone web with variation in consumer and top predator parameters (table 2b and figure 4). These results show that both the propensity to disperse and the strategies employed by dispersing species have profound influences on the species’ relative abundances in entire food webs embedded in a metacommunity.

Going beyond the evolution of dispersal propensity and community assembly, matching demographic habitat selection plays a potentially critical role in evolutionary processes [27,66], and modelling suggests that habitat selection successfully matching phenotypes to environments (directed gene flow) has greater adaptive potential than either adaptive plasticity or divergent natural selection in generating local adaptation and preventing local maladaptation [67]. A topic of future inquiry would be exploring the evolutionary implications of the ecological phenomena we have documented here .

The abundance inflation effect caused by dispersal, given spatiotemporal fitness variation that is highlighted in the present study, differs from other mass effects in that abundance inflation occurs when a species is regulated to the same average abundances in the various local communities. The magnitude of the inflation should increase with both the degree of temporal variation in abundance being experienced and when individuals disperse at higher rates, at least compared to very low dispersal (equation B4). Given that spatiotemporal variation in fitness favours higher dispersal propensities [14,15,17,28], the evolution of dispersal propensity should reinforce this inflation.

As noted above, in single-species metapopulations, temporal variation in local population growth rates can inflate time-averaged abundance [37–39]. This effect can be modulated by interspecific interactions. We have shown that abundance inflation in one species also distorts the abundances of other species with which it interacts in the food web. Not surprisingly, these distortions propagate as one would expect through the food web. Inflation of species at the bottom of the food web inflates the abundances of species above them or shifts the relative abundances among resource competitors, because more food is available for consumers [68]. This may even permit species to coexist that would otherwise not be able to support a population (e.g. the predator in the food chains with $N_2$ as the intermediate consumer in table 2). Likewise, inflation of species at the top of the food web exacerbates top-down effects and can shift the relative abundances among apparent competitors. Changes in top-down versus bottom-up control can also differentially affect dispersal propensities, which may also propagate through food webs [69]. Because of this propagation, identifying which species are the cause of the distortions will be difficult, but the strategies that species use to implement dispersal clearly play a critical role in the dynamics of metacommunities and whether or not dispersal alters average abundances beyond what is expected in the absence of dispersal.

We assumed that there is no direct cost of dispersal. Introducing such a cost does not alter the qualitative patterns we have presented, but, unsurprisingly, does reduce the overall level of dispersal at the evolutionary equilibrium (details not shown).

The evolution of dispersal need not lead to dominance by a single dispersal strategy. Earlier studies of dispersal evolution in single species [14,17,30] have shown that a polymorphism of robustly coexisting dispersal rates (one high, the other low) may be the evolutionarily stable strategy. The results we present here reveal that this can also arise in communities of interacting species, both for passively dispersing species and species with more active habitat selection. This mode of coexistence complements more classical ideas about coexistence arising because of competition–colonization trade-offs. In our models, all individuals are equivalent within patches in terms of their experience of fluctuations in density-independent growth factors, and intra- and interspecific interactions—i.e. no trade-off with dispersal propensity. Amarasekare [70] explored a model of a tri-trophic interaction where the intermediate consumer could passively disperse, and did not experience temporal variation directly. However, the top predator could go extinct in particular patches, then recolonize, leading to transient phases of fluctuations in abundance across the food chain, and thus in the intermediate consumer’s fitness. This emergent phase of unstable dynamics could promote the evolution of dispersal in the intermediate consumer. If the productivity of the basal species is spatially homogeneous, a single rate of passive dispersal is favoured. If instead the basal species vary greatly in productivity across patches, a dispersal polymorphism could be maintained. That paper demonstrated the importance of indirect interactions in modulating the evolution of dispersal, including the emergence of polymorphic, passive dispersal strategies; its results are consistent with our own (though note we introduce variation in fitness not via extinction, but directly in a density-independent growth parameter). For all of the questions above, and more, targeted experimental studies are needed, and because habitat selection is an animal phenomenon, these will be particularly challenging.

Future studies should explore a wider panoply of temporally varying processes, including covariation in density-independent growth factors, temporal autocorrelation and temporal variation directly in the strength of density dependence or the magnitude of interspecific interactions. Model (1) phenomenologically describes a food web where the constituent species have discrete generations; future analyses should examine more complex life histories, with, for instance, dispersal propensity varying with age or stage (which is known to be important, see [71]) and alternative food web modules. It is known that dispersal influences the stability of interacting species in metacommunities [72], and examining how stability and persistence are altered by dispersal evolution would be important to address. In addition to spatiotemporal fitness variation, future studies of dispersal evolution in a community context should also include interactions among kin [62]. Finally, additional behavioural rules for dispersal should be considered, as well as more realistic landscapes with dispersal localized to nearby patches, or mean growth parameters varying in more complex ways (e.g. along gradients or with temporal autocorrelation). We have focused on a simplified system and considered only two dispersal strategies, truly passive dispersal versus adaptive habitat selection: i.e. species that make no decision about immigration and those that always make particular kinds of adaptive decisions. In nature, coexisting species exist all along this continuum, and any given community/metacommunity is comprised species that vary not only in dispersal propensity but also in both their propensity/ability to select habitats based on their quality, and the factors and specific algorithms used in assessing habitat quality after emigration. We expect that the striking differences we have shown between passive dispersal and active habitat selection, in terms of dispersal evolution itself and its consequences for average abundances across a web of interacting species, will be maintained, and perhaps amplified, by including multiple sources of variation, but likely new surprises will emerge.

Appendix A. Optimal passive dispersal propensity when all patches in the metacommunity experience the same distribution of fitness variation

In this appendix, we derive the optimal passive dispersal propensity for a generalized conception of fitness variation among a set of n patches for a single species. Before dispersal, individuals of this species are distributed among the patches at some set of frequencies $f_{\left(g\right)}\left(t\right)$ , which is the proportion of individuals of this species that are in patch g at time t. For passive dispersal, a fraction $\epsilon$ of individuals in each patch emigrate and are equally apportioned among the other n − 1 patches. For this analysis, all dispersers are assumed to survive during dispersal.

All individuals in a patch then receive the same fitness value. The fitnesses that individuals obtain in a patch ${\displaystyle (W_{\left(g\right)}\left(t\right))}$ is a random variable. We assume that the fitnesses in all patches are drawn from the same distribution with mean $\overline{W}$ and variance ${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}\left(W\right)}$ . Fitnesses are also assumed to be uncorrelated in space and time (i.e. the fitness in each patch at each time is an independent random draw from this distribution).

Given these assumptions, within a generation, the mean fitness for the species is the spatial average of the fitnesses received by all individuals across all patches and is given by:

$${\displaystyle {\displaystyle \overline{W\left(t\right)}=\sum _{g=1}^n\left(\left(1-\epsilon \right)f_{\left(g\right)}\left(t\right)W_{\left(g\right)}\left(t\right)+\sum _{\begin{array}{c}i=1\\ \left(i\ne g\right)\end{array}}^p\epsilon \frac{f_{\left(g\right)}\left(t\right)}{n-1}{W}_{\left(i\right)}\left(t\right)\right),}}$$ (A 1)

where $f_{\left(g\right)}\left(t\right)$ is the frequency of individuals of the species in patch g before dispersal. Because the expected fitness is identical across all patches, the expected value of this quantity is ${\displaystyle E\left(\overline{W\left(t\right)}\right)=\overline{W}}$ , regardless of how individuals of the species are distributed across the patches.

Natural selection on dispersal propensity acts to maximize the long-term geometric mean fitness across generations [74,75]. Because the expected fitness is identical across all patches and thus also through time, the dispersal propensity that natural selection favours will be the $\epsilon$ that minimizes the variance in fitness across generations. The variance of this quantity is then given by

$${\displaystyle {\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}\left(\overline{W\left(t\right)}\right)=\mathrm{V}\mathrm{a}\mathrm{r}\left(W\right)\sum _{g=1}^p{\left(\left(1-\epsilon \right)f_{\left(g\right)}\left(t\right)+\sum _{\begin{array}{c}i=1\\ \left(i\ne g\right)\end{array}}^p\epsilon \frac{f_{\left(g\right)}\left(t\right)}{n-1}\right)}^2,}}$$ (A 2)

since spatial covariance in fitness is assumed to be zero, and so all covariance terms between fitnesses across patches are assumed to be zero. This is the variance in mean fitness through time, and natural selection on dispersal propensity will act to minimize this quantity.

Expanding this, the variance can be written as

$${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}\left(\overline{W\left(t\right)}\right)=\mathrm{V}\mathrm{a}\mathrm{r}\left(W\right)\left(\mathrm{\Lambda }+\mathrm{\Lambda }{\epsilon }^2-2\mathrm{\Lambda }\epsilon +\frac{{\epsilon }^2}{\left(n-1\right)}\mathrm{\Lambda }+\frac{4\epsilon }{\left(n-1\right)}\mathrm{\Phi }-\frac{4{\epsilon }^2}{\left(n-1\right)}\mathrm{\Phi }+\frac{2\left(n-2\right){\epsilon }^2}{{\left(n-1\right)}^2}\mathrm{\Phi }\right),}$$ (A 3)

where $\Lambda ={\displaystyle \sum _{g=1}^p{f}_{\left(g\right)}^2}$ and $\Phi ={\displaystyle \sum _{g=1}^p{f}_{\left(g\right)}{\displaystyle \sum _{\begin{array}{c}i=1\\ \left(i\ne g\right)\end{array}}^p{f}_{\left(i\right)}}}$ . Minimizing the variation in fitness across generations means finding the dispersal propensity $\epsilon$ that minimizes the quantity in parentheses in equation (A1.3). Setting the derivative of this quantity with respect to $\epsilon$ equal to 0 and solving for $\epsilon$ results in

$${\displaystyle \begin{array}{l}\epsilon =\frac{2\mathrm{\Lambda }-\frac{4}{n-1}\mathrm{\Phi }}{\frac{2n}{n-1}\mathrm{\Lambda }-\frac{4n}{{\left(n-1\right)}^2}\mathrm{\Phi }}\\ \epsilon =\frac{1}{\frac{n}{n-1}}\frac{2\mathrm{\Lambda }-\frac{4}{n-1}\mathrm{\Phi }}{2\mathrm{\Lambda }-\frac{4}{n-1}\mathrm{\Phi }}\\ \epsilon =\frac{n-1}{n}\end{array}.}$$ (A 4)

Thus, the passive dispersal propensity favoured by natural selection in a spatiotemporally varying environment in which all patches experience the same fitness variation is $\epsilon =\left(n-1\right)/n$ . The consequence of such dispersal is to equalize the abundances of the species across all patches each generation, which minimizes the variance in mean fitness through time.

To confirm that these general analytical results apply to a specific model of population regulation defining the fitnesses obtained by individuals of the species in the various patches, we simulated the model described in the main text with only the basal resource species, R, present in metacommunities having from 2 to 30 patches. When R passively dispersed among patches, the dispersal propensity evolved to match the analytical results here (i.e. equation A4) (figure 5).

Figure 5.

The dispersal propensity that evolves for R dispersing via passive dispersal or habitat selection in a metacommunity with 2 to 30 patches. The model presented in the main text with only R present is simulated for 50 000 iterations. The solid curve presents the dispersal propensity that evolved with passive dispersal, and the dashed line presents the dispersal propensity that evolved with habitat selection. The parameters are as given in Figure 1, with the values for each patch drawn from a normal distribution with mean 2.0 and standard deviation of 0.5.

Although we do not have analytical results, for comparison we also simulated R moving among patches via the habitat selection strategy. With habitat selection, a higher dispersal propensity evolved in metacommunities with more patches (figure 5). However, the dispersal propensity that evolved with habitat selection was lower than with passive dispersal.

Appendix B. A single-species continuous time model with passive dispersal

Before considering more complex multispecies scenarios, it is useful to keep in mind the population-level consequences for a single species dispersing among multiple patches, in each of which there is localized direct density dependence. This provides a conceptual yardstick for assessing the consequences of dispersal in a spatiotemporally varying environment.

Consider an environment consisting of p patches occupied by a single species, with local abundances of $R_{\left(g\right)}$ (the parenthetical g subscript denotes the patch, $g=1,2,3,\mathrm{...},p$ ). The species experiences logistic growth in each patch with the intrinsic rate of increase of $c_{\left(g\right)}$ in patch g and a strength of density dependence $d$ [76,77]. We assume the intrinsic rate of increase $c_{\left(g\right)}\left(t\right)$ varies with time and is a normally distributed stochastic random variable (uncorrelated in time and space), and we assume the strength of density dependence is spatially and temporally invariant. A constant fraction $\epsilon$ of the population in each patch emigrates, and dispersers are distributed equally across the other patches. Migrants experience mortality in transit; a fraction s of emigrants survive their dispersal journey. For simplicity, in this section, we assume continuous time. These assumptions lead to the following metapopulation model:

$${\displaystyle {\displaystyle \frac{1}{R_{\left(g\right)}}\frac{d{R}_{\left(g\right)}}{dt}=c_{\left(g\right)}\left(t\right)-d{R}_{\left(g\right)}-\epsilon +s\epsilon \frac{\sum _{j=1\left(g\ne j\right)}^n{R}_{\left(j\right)}}{\left(p-1\right)R_{\left(g\right)}}.}}$$ (B 1)

With no temporal variation (i.e. $c_{\left(g\right)}\left(t\right)=c_{\left(g\right)}$ ) and no dispersal (i.e. $\epsilon =0$ ), the equilibrium abundance in each patch is

$${\displaystyle R_{\left(g\right)}^{\ast }=\frac{c_{\left(g\right)}}{d}.}$$ (B 2)

Given dispersal (i.e. ${\displaystyle \epsilon >0}$ ) but without temporal variation in $c_{\left(g\right)}\left(t\right)$ , the equilibria in the various patches become more similar as dispersal propensity increases [16]; patches with higher $R_{\left(g\right)}^{*}$ are net exporters of migrants and patches with lower ${\displaystyle R_{\left(g\right)}^{\ast }}$ are net importers.

If patches differ in their intrinsic equilibrial abundances (i.e. $c_{\left(g\right)}/d\ne c_{\left(j\right)}/d$ for some combination of patches), dispersal can cause the total number of individuals in the metapopulation to differ from ${\displaystyle \sum _{g=1}^p{c}_{\left(g\right)}/d}$ ―in other words, if patches are regulated to different equilibrium abundances, dispersal distorts total abundance of the entire metapopulation [16,56,57,60]. For example, for two patches in equation (A1), the total abundance is

$${\displaystyle R_{\left(1\right)}^{\ast }+R_{\left(2\right)}^{\ast }=\frac{c_{\left(1\right)}}{d}+\frac{c_{\left(2\right)}}{d}+\epsilon \left(\frac{s\left({\left(R_{\left(1\right)}^{\ast }\right)}^2+{\left(R_{\left(2\right)}^{\ast }\right)}^2\right)-2R_{\left(1\right)}^{\ast }{R}_{\left(2\right)}^{\ast }}{d{R}_{\left(1\right)}^{\ast }{R}_{\left(2\right)}^{\ast }}\right).}$$ (B 3)

The last term in equation 2.3 determines the deviation in total abundance in the metapopulation from that determined by local population regulation (i.e. $c_{\left(1\right)}/d+c_{\left(2\right)}/d$ ). This deviation increases with dispersal propensity (i.e. as $\epsilon$ increases) and with increasing differences in local abundances among patches: increasing difference in local abundances increases the numerator of the last term in equation (B3) (see also [16,56–58,60]). However, if ecological conditions in all patches imply equal abundances (i.e. ${\displaystyle R_{\left(g\right)}^{\ast }=R^{\ast }}$ for all patches) and $s=1$ , total metapopulation abundance will not deviate from ${\displaystyle \sum _{g=1}^p{c}_{\left(g\right)}/d}$ , because the numerator of the last term of equation (B3) is zero. The deviation is positive if patches differ primarily in intrinsic rates of increase [56,78], but the deviation can be negative if patches differ primarily in the strength of density dependence (i.e. variation among patches in $d$ ) [56] or disperser survival is low (i.e. the first term in the numerator of the last term of equation (B3) is small because of a small value of s). If some habitats are sinks with ${\displaystyle c_{\left(g\right)}<0}$ , dispersal can boost local abundance above that expected with no dispersal [16].

These effects of dispersal on total abundance occur in temporally static environments. An additional mechanism arises given spatiotemporal variation in intrinsic growth rates [38,51]. Consider the long-term average per capita population growth rate of each population, ${\displaystyle \overline{\frac{1}{R_{\left(g\right)}}\frac{d{R}_{\left(g\right)}}{dt}}}$ (the overbar signifies the time-averaged arithmetic mean), which changes because of dispersal and spatiotemporal environmental variation in $c_{\left(g\right)}\left(t\right)$ .

To find the expected long-term average per capita fitness of this dispersal type, we find the quadratic approximation of the long-term average by taking the multivariable Taylor series expansion of equation (B1) at the long-term average abundances of the dispersal type in each patch, and retaining all constant, linear and quadratic terms [26,79,80]. Because we assume that the intrinsic rate of increase $c_{\left(g\right)}\left(t\right)$ varies in an uncorrelated fashion in time and space, $c_{\left(g\right)}\left(t\right)$ and the abundances of this dispersal type in all patches of the metacommunity are random variables in equation (B1).

Equating $\frac{1}{R_{\left(g\right)}}\frac{d{R}_{\left(g\right)}}{dt}=f\left(c_{\left(g\right)}\left(t\right),R_{\left(1\right)},R_{\left(2\right)},\mathrm{...},R_{\left(g\right)},\mathrm{...},R_{\left(p\right)}\right)$ , the Taylor series expansion of equation (2.1) to the second order has the following form:

$$\begin{gathered} {\displaystyle {\displaystyle f\left(c_{(g)}(t),R_{(1)},R_{(2)},...,R_{(g)},...,R_{(P)}\right)\approx f\left({\overline{c}}_{(g)},{\overline{R}}_{(1)},{\overline{R}}_{(2)},...,{\overline{R}}_{(g)},...,{\overline{R}}_{(P)}\right)} \\ +\frac{\mathrm{\partial }(c_{(g)}(t),R_{(1)},R_{(2)},...,R_{(g)},...,R_{(P)}}{\mathrm{\partial }{c}_{(g)}(t)}\left(c_{(g)}(t)-{\overline{c}}_{(g)}\right) \\ +\sum _{i=1}^p\frac{\mathrm{\partial }f(c_{(g)}(t),R_{(1)},R_{(2)},...,R_{(g)},...,R_{(P)})}{\mathrm{\partial }{R}_{(i)}}\left(R_{(i)}-{\overline{R}}_{(i)}\right) \\ +\sum _{i=1}^p\frac{{\mathrm{\partial }}^{2}f(c_{(g)}(t),R_{(1)},R_{(2)},...,R_{(g)},...,R_{(P)})}{\mathrm{\partial }{c}_{(g)}(t)\mathrm{\partial }{R}_{(i)}}\left(c_{(g)}(t)-{\overline{c}}_{(g)}\right)(R_{(i)}-{\overline{R}}_{(i)}) \\ +\sum _{i=1}^p\sum _{j=1}^p\frac{{\mathrm{\partial }}^{2}f\left(c_{(g)}(t),R_{(1)},R_{(2)},...,R_{(g)},...,R_{(P)}\right)}{\mathrm{\partial }{R}_{(i)}\mathrm{\partial }{R}_{(j)}}(R_{(i)}-{\overline{R}}_{(i)})(R_{(j)}-{\overline{R}}_{(j)})} \end{gathered}$$

Because the expected values of all the linear terms are zero, these vanish from the expectation. Also, we equate the expected values of the quadratic terms with the associated variances and covariances among the random variables [26]. A number of these terms also vanish because the second derivatives of $\frac{1}{R_{\left(g\right)}}\frac{d{R}_{\left(g\right)}}{dt}$ with respect to the two associated variables are zero.

From the Taylor series expansion to the per capita population growth rates, the long-term average per capita growth rate of each population is seen to be a function of both the local variation in population size and covariances of population sizes among all pairs of patches:

$${\displaystyle {\displaystyle \begin{array}{l}\overline{\frac{1}{R_{\left(g\right)}}\frac{d{R}_{\left(g\right)}}{dt}}\approx \overline{c_{\left(g\right)}\left(t\right)}-d\overline{R_{\left(g\right)}}-\epsilon +s\epsilon \frac{\sum _{j=1\left(g\ne j\right)}^n\overline{R_{\left(j\right)}}}{\left(p-1\right)\overline{R_{\left(g\right)}}}\\ +\frac{s\epsilon }{{\overline{R_{\left(g\right)}}}^3}\left(\frac{\sum _{j=1\left(g\ne j\right)}^n\overline{R_{\left(j\right)}}}{p-1}\mathrm{V}\mathrm{a}\mathrm{r}\left(R_{\left(g\right)}\right)-\overline{R_{\left(g\right)}}\sum _{j=1\left(g\ne j\right)}^p\mathrm{C}\mathrm{o}\mathrm{v}\left(R_{\left(g\right)},R_{\left(j\right)}\right)\right)\end{array}.}}$$ (B 4)

Equation (A4) illustrates that the average long-term per capita population growth rate for a given combination of abundances in each patch is (i) increased by temporal variation in local abundance; (ii) increased or decreased by the temporal covariation in abundance between patches (depending on the sign of the covariation); and (iii) increased by the dispersal propensity $\epsilon$ if variances and covariances are not zero (at high s). Thus, the long-term average abundance should increase or decrease in accordance with these changes. For example, if the average long-term per capita growth rate increases at all abundance combinations, the average long-term population abundance should also increase. Note that with no dispersal (i.e. $\epsilon =0$ ), spatiotemporal variation per se does not alter the spatial pattern of average abundances (i.e. $\overline{R_{\left(g\right)}}=\overline{c_{\left(g\right)}\left(t\right)}/d$ ). Intriguingly, the long-term growth rates in each population can be positive, even with all ${\displaystyle \overline{c_{\left(g\right)}\left(t\right)}/d<t0}$ because of this supplementation due to dispersal (as in [38], who focused exclusively on sink habitats, and in a complementary manner pointed out the importance of covariance between local densities and local growth rates).

These equations are complex and calculating the long-term average population size in each patch directly is not analytically tractable. However, we can explore these changes graphically by considering how isoclines for dynamics in the patches are changed by the interplay of dispersal and spatiotemporal variation. Figure 6 illustrates the isoclines for the model with two patches (i.e. p = 2). Simulations suggest that spatiotemporal variation in $c_{\left(g\right)}\left(t\right)$ typically generates ${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}\left(R_{\left(g\right)}\right)\ge \mathrm{C}\mathrm{o}\mathrm{v}\left(R_{\left(g\right)},R_{\left(j\right)}\right)}$ , and so the long-term average population sizes in patches may typically exceed that seen without such variation. As noted above, in the absence of spatiotemporal variation, dispersal does not distort population sizes away from those caused by local ecological conditions if the intrinsic rates of increase and strengths of density dependence are the same in both patches (note the point of intersection between the isoclines of solid curves in figure 6a ). The addition of spatiotemporal variation inflates the long-term average abundances in both patches (the point of intersection between the isoclines of large-dashed curves in figure 6). This abundance inflation is distinct from the effect discussed earlier (equation B3) because ${\displaystyle \overline{c_{\left(1\right)}\left(t\right)}/d=\overline{c_{\left(2\right)}\left(t\right)}/d}$ .

Figure 6.

Phase space plots of the isoclines from equation A4 for a metapopulation of two patches occupied by a single species. The two thin-dashed lines are the isoclines for each patch with no dispersal of individuals between the two patches. The two solid curves are the isoclines for each patch with dispersal but no temporal variation in c _i (t) (intrinsic rate of increase). The two large-dashed curves are the corresponding isoclines for the two patches with dispersal between the patches and with temporal variation in c _i (t). The long-term expected population sizes in the two patches are the points where the two corresponding isoclines intersect. Panel (a) shows the conditions when the average intrinsic rates of increase in the two patches are the same, and panel (b) shows the conditions when the average intrinsic rate of increase in patch 1 is twice the value of that in patch 2. Other parameter values used to make the two panels are d1 = d2 = 0.2, and c _i (t) drawn from a normal distribution with specified mean (shown above panel) and standard deviations of 0.75.

In contrast, when the intrinsic rates of increase of the two patches differ but do not vary with time, dispersal causes the equilibrium abundances in the patches to converge [16]: namely, if ${\displaystyle c_{\left(1\right)}>c_{\left(2\right)}}$ , the equilibrium abundance in patch 1 will be lower and in patch 2 will be higher with dispersal than local ecological conditions favour (note the point of intersection of the solid curves as compared with the point of intersection of the two thin-dashed lines in figure 6b ). This equilibrium includes the excess of individuals above ${\displaystyle \sum _{g=1}^p{c}_{\left(g\right)}/d}$ generated by the mechanisms sketched above in equation (B3) [16,56,57,60]. With temporal variation in both patches, the long-term average abundances of both patches are again inflated, but the abundance in the patch with the lower average intrinsic rate of increase is proportionally greater (compare the point of intersection of the thick-dashed curves to the point of intersection of the solid curves in figure 6). These same inflation patterns hold with more patches ( ${\displaystyle p>t2}$ ), namely, the abundances in all patches are inflated, compared to no temporal variation, but patches that support fewer individuals are inflated proportionally more by dispersal and spatiotemporal variation.

Ethics

This work did not require ethical approval from a human subject or animal welfare committee.

Data accessibility

Electronic supplementary material is available online [73].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

M.M.: conceptualization, formal analysis, software, validation, visualization, writing—original draft, writing—review and editing; W.R.: conceptualization, resources, supervision, validation, visualization, writing—original draft, writing—review and editing; R.D.H.: investigation, validation, visualization, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

R.D.H. thanks the University of Florida Foundation and NSF awards DEB 1923495 and DEB 1923513 for their support.